A linear preserver problem on maps which are triple derivable at orthogonal pairs

Abstract

A linear mapping T on a JB∗-triple is called triple derivable at orthogonal pairs if for every a,b,c∈E with a⊥b we have 0={T(a),b,c}+{a,T(b),c}+{a,b,T(c)}. We prove that for each bounded linear mapping T on a JB∗-algebra A the following assertions are equivalent: (a) T is triple derivable at zero; (b) T is triple derivable at orthogonal elements; (c) There exists a Jordan ∗-derivation D:A→A∗∗, a central element ξ∈A∗∗sa, and an anti-symmetric element η in the multiplier algebra of A, such that T(a)=D(a)+ξ∘a+η∘a, for all a∈A;

(d) There exist a triple derivation δ:A→A∗∗ and a symmetric element S in the centroid of A∗∗ such that T=δ+S. The result is new even in the case of C∗-algebras. We next establish a new characterization of those linear maps on a JBW∗-triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW∗-triple M, the following statements are equivalent for each bounded linear mapping T on M: (a) T is triple derivable at orthogonal pairs; (b) There exists a triple derivation δ:M→M and an operator S in the centroid of M such that T=δ+S. \end{enumerate}